This fraction must now be reduced to a decimal on the principles of the last article, by the rule usually given, either exactly, or by approximation, according to the nature of the factors in the denominator. When the decimal fraction corresponding to a common fraction cannot be exactly found, it always happens that the series of decimals which approximates to it, contains the same number repeated again and again. Thus, in the example which we chose, is more and more nearly represented by the fractions .6, .63, .636, .6363, &c., and if we carried the process on without end, we should find a decimal fraction consisting entirely of repetitions of the figures 63 after the decimal point. Thus, in finding, the figures which are repeated in the numerator are 142857. This is what is commonly called a circulating decimal, and rules are given in books of arithmetic for reducing them to common fractions. We would recommend to the beginner to omit all notice of these fractions, as they are of no practical use, and cannot be thoroughly understood without some knowledge of algebra. It is sufficient for the student to know, that he can always either reduce a common fraction to a decimal, or find a decimal near enough to it for his purpose, though the calculation in which he is engaged requires a degree of accuracy which the finest microscope will not appreciate. But in using approximate decimals there is one remark of importance, the necessity for which occurs continually. Suppose that the fraction 2.143876 has been obtained, and that it is more than sufficiently accurate for the calculation in which it is to be employed. Suppose that for the object proposed it is enough that each quantity employed should be a decimal fraction of three places only, the quantity 2.143876 is made up of 2.143, as far as three places of decimals are concerned, which at first sight might appear to be what we ought to use, instead of 2.143876. But this is not the number which will in this case give the utmost accuracy which three places of decimals will admit of; the common usages of life will guide us in this case. Suppose a regiment consists of 876 men, we should express this in what we call round numbers, which in this case would be done by saying how many hundred men there are, leaving out of consideration the number 76, which is not so great as 100; but in doing this we shall be nearer the truth if we say that the regiment consists of 900 men instead of 800, because 900 is nearer to 876 than 800. In the same manner, it will be nearer the truth to write 2.144 instead of 2.143, if we wish to express 2.143876 as nearly as possible by three places of decimals, since it will be found by subtraction that the first of these is nearer to the third than the second. Had the fraction been 2.14326, it would have been best expressed in 2.1435, it would have been equally well three places by 2.143; had it been expressed either by 2.143 or 2.144, both being equally near the truth; but 2.14351 is a little more nearly expressed by 2.144 than by 2.143. We have now gone through the leading principles of arithmetical calculation, considered as a part of general Mathematics. With respect to the commercial rules, usually considered as the grand object of an arithmetical education, it is not within the scope of this treatise to enter upon their consideration. The mathematical student, if he is sufficiently well versed in their routine for the purposes of common life, may postpone their consideration until he shall have become familiar with algebraical operations, when he will find no difficulty in understanding the principles or practice of any of them. He should, before commencing the study of algebra, carefully review what he has learnt in arithmetic, particularly the reasonings which he has met with, and the use of the signs which have been introduced. Algebra is at first only arithmetic under another name, and with more general symbols, nor will any reasoning be presented to the student which he has not already met with in establishing the rules of arithmetic. His progress in the former science depends most materially, if not altogether, upon the manner in which he has attended to the latter; on which account the detail into which we have entered on some things which to an intelligent person are almost self-evident, must not be deemed superfluous. When the student is well acquainted with the principles and practice of arithmetic, and not before, he should commence the study of algebra. It is usual to begin algebra and geometry together, and if the student has sufficient time, it is the best plan which he can adopt. Indeed, we see no reason why the elements of geometry should not precede C those of algebra, and be studied together with arithmetic. In this case the student should read that part of these treatises which relates to geometry, first. It is hardly necessary to say that though we have adopted one particular order, yet the student may reverse or alter that order so as to suit the arrangement of his own studies. We now proceed to the first principles of algebra, and the elucidation of the difficulties which are found from experience to be most perplexing to the beginner. We suppose him to be well acquainted with what has been previously laid down in this treatise, particularly with the meaning of the signs +, -, ×, and the sign of division. CHAPTER VI. Algebraical Notation and Principles. WHENEVER any idea is constantly recurring, the best thing which can be done for the perfection of language, and consequent advancement of knowledge, is to shorten as much as possible the sign which is used to stand for that idea. All that we have accomplished hitherto has been owing to the short and expressive language which we have used to represent numbers, and the opera tions which are performed upon them. The first step was to write simple signs for the first numbers, instead of words at full length, such as 8 and 7, instead of eight and seven. The next was to give these signs an additional meaning, according to the manner in which they were connected with one another; thus 187 was made to represent one hundred added to eight tens added to seven. The First No+second N° First 2 + next was to give by new signs directions when to perform the operations of addition, subtraction, multiplication, and division; thus 5+8 was made to represent 8 added to 5, and so on. With these signs reasonings were made, and truths discovered which are common to all numbers; not at once for every number, but by taking some example, by reasoning upon it, and by producing a result; this result led to a rule which was declared to be a rule which held equally good for all numbers, because the reasoning which produced it might have been applied to any other example as well as to the one which was chosen. In this way we produced some results, and might have produced many more; the following is an instance-half the sum of two numbers added to half their difference, gives the greater of the two numbers. For example, take 16 and 10, half their sum is 13, half their difference is 3; if we add 13 and 3 we get 16, the greater of the two numbers. We might satisfy ourselves of the truth of this same proposition for any other numbers, such as 27 and 8, 15 and 19, and If we then make use of signs, we find the following truths :16+10 16-10 so on. 27+8 27-8 + : 16. 2 2 + = 27. 2 15+9 2 + 2 15-9 2 = 15, and so on. In this way we might express anything suppose the first N°, the second N°, &c. = When any sentence expresses that two numbers or collections of numbers are equal to one another, it is called an equation, thus 7+5=12 is an equation, and the sentences which have been just written down are equations. Now the next question is, could we not avoid the trouble of writing first N°, second N°, &c. so frequently? This is done by putting letters of the alphabet to stand for these numbers. Suppose, for example, we let a stand for the first number, and y for the second, the two assertions already made will then be written thus: to stand for numbers, and whenever a letter is used it means either that any number may be used instead of that letter, or that the number which the letter stands for is not known, and that the letter supplies its place in all the reasonings until it is known. 2. The sign is used for addition, as in arithmetic. Thus +z is the sum of the numbers represented by x and z. The following equations are sufficiently evident : x+y+z = x+z+y=y+z+x. If a = b, then a+c=b+c, a +c+d=b+c+d, &c. 3. The sign is used for subtraction, as in arithmetic. The following equations will show its use : x+α- b c+e=x+a+e b-ca-c+e b+x. If ab, a-cbc, a-c+d=b-c+d, &c. 4. The sign x is used for multiplication as in arithmetic, but when two numbers represented by letters are multiplied together it is useless, since axb can be represented by putting a and b together thus, a b. Also a xbx c is represented by a b c ;" a × a × a, for the present we represent thus, a a a. When two numbers are multiplied together, it is necessary to keep the sign x; otherwise 7 x 5 or 35 would be mistaken for 75. It is, however, usual to place a point between two numbers which are to be multiplied together; thus 7 × 5 × 3 is written 7.5.3. But this point may sometimes be mistaken for the decimal point: this will, however, be avoided by always writing the decimal point at the head of the figure, viz. by writing 23461 thus, 234.61. α 100 repre 5. Division is written, as in arithmetic; thus, signifies that the number b sented by a is to be divided by the number represented by b. 6. All collections of numbers are called expressions; thus, a+b, a+b-c, aa+bb-d, are algebraical expressions. ХХХ 7. When two expressions are to be multiplied together, it is indicated by placing them side by side, and inclosing each of them in brackets. Thus, if a+b+c is to be multiplied by d+e+f, the product is written in any of the following ways:— (a+b+c) (d+e+f), a+b+c.d+e+f, thus, a >b. thus, a < b. 10. When there is a product in which all the factors are the same, such as xxxxx,which means that five equal numbers, each of which is represented by x, are multiplied together, the letter is only written once, and above it is written the number of times which it occurs, thus, xxxxx is written a The following table should be carefully studied by the student : or xx is written a2, and is called the square, or second power of x• There is no point which is so likely to create confusion in the ideas of a beginner as the likeness between such expressions as 4x and x. On this account it would be better for him to omit using the latter expression, and to put xxxx in its place until he has acquired some little facility in the operations of algebra. If he does not pursue this course he must recollect that the 4, in these two expressions, has different The difference of meaning will be appanames and meanings. In 4x it is called rent from the following tables :a coefficient, in x an exponent or index. The beginner should frequently write for himself such expressions as the following: 4a baaabb +aaabb+aaabb +aaabb 5а2 х=аааах + aaaax+aaaax+aaaax+ аааах ab α a2+b + α, I. when a+b -b aa stands for 6, and b for 5, II. when a stands for 13, and b for 2, and so on. He should stop here until he has, by these means, made the signs familiar to his eye and their meaning to his mind; nor should he proceed to any further algebraical operations until he can readily find the value of any algebraical expression, when he knows the numbers which the letters stand for. He cannot, at this period of his course, write too many algebraical expressions, and he must particularly avoid slurring over the sense of what he has before him, and must write and rewrite each expression bb = аа сс bb+cc + aa-bb aa-bb aa+ab+bb a+b until the meaning of the several parts forces itself upon his memory at first sight, without even the necessity of putting it in words. It is the neglecting to do this which renders the operations of algebra so tedious to the beginner. He usually proceeds to the addition, subtraction, &c. of symbols, of the meaning of which he has but an imperfect idea, and which have been newly introduced to him in such numbers that perpetual confusion is the consequence. We cannot, therefore, use too many arguments to induce him not to mind the drudgery of reducing algebraical expressions into figures. This is the connecting link between the new science and arithmetic, and, unless that link be well fastened, the knowledge which he has previously acquired in arithmetic will help him but little in acquiring algebra. The order of the terms of any algebraical expression may be changed without changing the value of that expression. This needs no proof, and the following are examples of the change :— a+b+ab+c+ac-d-e-de-f =a-d+be+ab-de+c-f+ac =a+b-d-e-de-f+ac+c+ab =ab+ac-de+a+b+c-e-f-d When the first term changes its place, as in the fourth of these expressions, the sign+ is put before it, and should, properly speaking, be written wherever there is no sign, to indicate that the term in question increases the result of the rest, but it is usually omitted. The negative sign is often written before the first term, as in the expression -a+b: but it must be recollected that this is written on the supposition that a is subtracted from what comes after it. When an expression is written in brackets, with some sign before it, such as a- (b-c), it is understood that the expression in brackets is to be considered as one quantity, and that its result or total is to be connected with the Similarly a-(b-c)=a-b+c rest by the sign which precedes the a+b-(c+d-e-f)=a+b-c-d+e+f (ax2—bx+c)—(dx2—ex+ƒ)=ax2-bx+c-dx2+ex—ƒ. When the positive sign is written not be reduced, which they do by adding before an expression in brackets, the brackets may be omitted altogether, unless they serve to show that the expression in question is multiplied by some other. Thus, instead of (a+b+c)+(d+e+f), we may write a+b+c+d+e+f, which is, in fact, only saying that two wholes may be added together by adding together all the parts of which they are composed. But the expression a+b+c) (d+e) must not be written thus: a+b+c (d+e), since the first expresses that (b+c) must be multiplied by (d+e) and the product added to a, and the second that c must be multiplied by (d+e) and the product added to a+b. If a, b, c, d, and e, stand for 1, 2, 3, 4, and 5, the first is 46 and the second 30. When two or more quantities have exactly the same letters repeated the same number of times, such as 4a b3, and 6a b3, they may be reduced into one by merely adding the coefficients and retaining the same letters. Thus, 2a+3a is 5a, 6bc-4bc is 2bc, 3 (x+y)+2(x+y) is 5 (x+y). These things are evident, but beginners are very liable to carry this farther than they ought, and to attempt to reduce expressions which do not admit of reduction. For example, they will say that 36+62 is 46 or 462, neither of which is true, except when b stands for 1. The expression 3b+b2, or 3b+bb, cannot be made more simple until we know what b stands for. The following table will, perhaps, be of service. the exponents of letters as well as their coefficients, or by collecting several terms into one, and leaving out the signs of addition and subtraction. The beginner cannot too often repeat to himself that two terms can never be made into one, unless both have the same letters, each letter being repeated the same number of times in both, that is, having the same index in both. When this is the case, the expressions may be reduced by adding or subtracting the coefficients according to the sign, and affixing the common letters with their indices. When there is no coefficient, as in the expression ab, the quantity represented by a2 b being only taken once, 1 is called the coefficient. Thus, = 3 ab + 4 ab +6 ab — ab — 7 ad = 5 ad 6 xy + 3 xy2 — 5 xy2 + xy2 5 xy2 The student must also recollect that he is not at liberty to change an index from changes the quantity represented. Thus one letter to another, as by so doing he ab and ab1 are quantities totally distinct, the first representing aaaab and the second abbbb. The difference in all the cases which we have mentioned will be made more clear, by placing numbers at pleasure instead of letters in the expres sions, and calculating their values; but, in conclusion, the following remark must be attended to. If it were asserted that a2 + b2 a + b and we wish to pro the expression a+b 2 ab is the same as ceed to see whether this is always the case or no, if we commence accidentally by supposing b to stand for 2 and a for 4, we shall find that the first is the same as the second, each being 33. But we must not conclude from this that they are always the same, at least until we |